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From: jrk@information-systems.east-anglia.ac.uk (Richard Kennaway CMP RA)
Newsgroups: comp.ai.philosophy
Subject: Re: Intuition and doubt (was Re: Minds, machines, and Godel)
Message-ID: <17928.9101281842@s4.sys.uea.ac.uk>
Date: 28 Jan 91 18:42:50 GMT
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In article <1991Jan25.075046.4464@sics.se> torkel@sics.se (Torkel Franzen)
writes:

>My question is, what proofs in elementary
>analysis, if any, do you regard as proving anything?

All of them - provided elementary analysis is consistent.

>If FLT turned out
>to be provable in analysis, would you still say that we have no reason
>for believing it to be true (i.e.  that the equation x^n+y^n=z^n has
>no non-trivial solutions for n>2) except that no counterexample has
>been found - as you do in the case of the theorem "elementary
>arithmetic is consistent"?

The reason - at least, my reason - for believing FLT under such
circumstances is not the "statistical" evidence of no counterexample to
FLT having been found, but of no counterexample to the consistency of
elementary analysis having been found.

...
>Hence my initial remarks on the merely ritual character of much talk about
>consistency.

Quite so.  Consistency is no more worth remarking on than that the earth
turns.  One does not usually bother to preface a mathematical proof with
"provided arithmetic (or analysis, ZF, etc.) is consistent".  It begins
to matter, when people start claiming (as in the discussion that this
subthread arose from) that arithmetic "really" is true, rather than
merely exhibiting a certain syntactic object (viz. a proof of consistency
of arithmetic in elementary analysis).  There is no problem with the
latter.  The former is - to me - unclear and unnecessary.

--
Richard Kennaway          SYS, University of East Anglia, Norwich, U.K.
Internet:  jrk@sys.uea.ac.uk            uucp:  ...mcsun!ukc!uea-sys!jrk